Is there a Rule of Differentiation which gives a closed form evaluation for the integral of an inverse function F^-1 in terms of F?

[Al-Layth]

Is there a Rule of Differentiation which gives a closed form evaluation for the integral of an inverse function in terms of the original function and its derivatives and antiderivatives?

[math]\frac{d}{dx}F^{-1}=?[/math]
As this would be very helpful to me right now )))

 

[Dr.Peterson]

Is there a Rule of Differentiation which gives a closed form evaluation for the integral of an inverse function in terms of the original function and its derivatives and antiderivatives?

[math]\frac{d}{dx}F^{-1}=?[/math]
As this would be very helpful to me right now )))

Do you mean something like the inverse function theorem?

3.7: Derivatives of Inverse Functions

The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. We can use the inverse function theorem to develop …

math.libretexts.org

I'm not sure what you mean about an integral.

 

[Al-Layth]

Do you mean something like the inverse function theorem?

3.7: Derivatives of Inverse Functions

The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. We can use the inverse function theorem to develop …

math.libretexts.org

I'm not sure what you mean about an integral.

Damn I brainfarted hard there. Twice. Sorry, please Ignore the use of “integral” . I would edit but I can’t figure out how.?

yes the inverse function theorem looks like exactly what I need. One more Q though just to confirm,
If I knew the derivative of sin(x) and I knew this inverse function theorem, and I did not know the derivative of arcsin(x) , could I deduce it with this theorem and my knowledge of the derivative of sin(x)?

 

[Dr.Peterson]

Damn I brainfarted hard there. Twice. Sorry, please Ignore the use of “integral” . I would edit but I can’t figure out how.?

yes the inverse function theorem looks like exactly what I need. One more Q though just to confirm,
If I knew the derivative of sin(x) and I knew this inverse function theorem, and I did not know the derivative of arcsin(x) , could I deduce it with this theorem and my knowledge of the derivative of sin(x)?

Give it a try! Yes, you should be able to. You'll find that it does take a little work (specifically trig).